In some computation I have to do, I have to deal with the following situation: I have an arbitrary number of tetrahedra (=3 simplexes) and I am allowed to identify pairs of faces to each other. There is no restriction in how this is done, for example it is also allowed to identify two faces, which belong to the same tetrahedra. Each face is only allowed to be identified once, so I it not possible to have three faces identified to each other. Furthermore, it is also possible that after this procedure there are some faces, which are not identified to anything else.
Now my question is the following: If i know the number of edges, faces and tetrahedra, as well as as the number of face identifications, is it possible to calculate the number of vertices? I naively guessed at the beginning the following formula $$4\text{ number of tetrahedra} - 3\text{ number of identifiactions}$$
in which i basically viewed every tetrahedron (which has 4 vertices) at the beginning to be disjoint and substract three vertices for each identification. However, this is obviously wrong, since when I glue 4 tetrahedra as in the following figure, the answer is clearly wrong:
Here my calculation gives $4\cdot 4- 3\cdot 6=-2$, which is quite far from the correct answer $5$.
Any idea how do it correctly?